Climbing up a random subgraph of the hypercube

Michael Anastos, Sahar Diskin, Dor Elboim, Michael Krivelevich

Research output: Contribution to journalArticlepeer-review

Abstract

Let Qd be the d-dimensional binary hypercube. We say that P = {v1, …, vk } is an increasing path of length k − 1 in Qd, if for every i ∈ [k − 1] the edge vivi+1 is obtained by switching some zero coordinate in vi to a one coordinate in vi+1. Form a random subgraph Qdp by retaining each edge in E(Qd) independently with probability p. We show that there is a phase transition with respect to the length of a longest increasing path around p =ed. Letαbe a constant and letp=αd. When 0 < α < e, then there exists a δ ∈ (0, 1) such that whp a longest increasing path in Qdp is of length at most (1 − δ)d. On the other hand, when α > e, whp there is a path of length d − 2 in Qdp, and in fact, whether it has length d − 2, d − 1, or d depends on whether the vertices (0, …, 0) and (1, …, 1) are in the giant connected component.

Original languageEnglish
Article number70
JournalElectronic Communications in Probability
Volume29
DOIs
StatePublished - 2024

Funding

FundersFunder number
Horizon 2020 Framework Programme
H2020 Marie Skłodowska-Curie Actions101034413

    Keywords

    • bond percolation
    • hypercube
    • phase transition

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